Schema_RK_Rationnel.cpp

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#include <Schema_RK_Rationnel.h>
#include <Equation_base.h>
 
// XD runge_kutta_rationnel_ordre_2 schema_temps_base runge_kutta_rationnel_ordre_2 INHERITS_BRACE This is the
// XD_CONT Runge-Kutta rational scheme of second order. The method is described in the note: Wambeck - Rational
// XD_CONT Runge-Kutta methods for solving systems of ordinary differential equations, at the link:
// XD_CONT https://link.springer.com/article/10.1007/BF02252381. Although rational methods require more computational
// XD_CONT work than linear ones, they can have some other properties, such as a stable behaviour with explicitness,
// XD_CONT which make them preferable. The CFD application of this RRK2 scheme is described in the note:
// XD_CONT https://link.springer.com/content/pdf/10.1007\%2F3-540-13917-6_112.pdf.
Implemente_instanciable(RRK2,"Runge_Kutta_Rationnel_ordre_2",TRUSTSchema_RK<Ordre_RK::RATIO_DEUX>);
 
Sortie& RRK2::printOn(Sortie& s) const { return  TRUSTSchema_RK<Ordre_RK::RATIO_DEUX>::printOn(s); }
 
Entree& RRK2::readOn(Entree& s) { return TRUSTSchema_RK<Ordre_RK::RATIO_DEUX>::readOn(s) ; }
 
/*! @brief Performs a second-order rational Runge-Kutta time step on the equation passed as parameter.
 *
 * The second-order rational Runge-Kutta scheme:
 *      g1=hf(y0)
 *      g2=hf(y0+c2g1)
 *      y1=y0+(g1g1)/(b1g1+b2g2)
 *      where ab/d = (a(b,d)+b(d,a)-d(a,b)/(d,d)
 *      y1=y0+(2g1(g1,b1g1+b2g2)-(b1g1+b2g2)(g1,g1)/(b1g1+b2g2,b1g1+b2g2)
 *      y1=y0+(2g1(g1,"g2")-("g2")(g1,g1)/("g2","g2")
 *       order 2 if b2c2=-1/2
 *      b2c2<=-1/2 A0 stability and I stability
 *      b2c2<= 1/(2cos(alpha)(2-cos(alpha))) 0<=alpha<pi/2 Aalpha stability
 *
 */
int RRK2::faire_un_pas_de_temps_eqn_base(Equation_base& eqn)
{
  // Warning on the first 100 time steps if facsec equals 1, to prompt the user to reconsider
  if (nb_pas_dt() >= 0 && nb_pas_dt() <= NW && facsec_ == 1) print_warning(NW);
 
  const double b1 = 2.0, b2 = -1, c2 = 0.5;
 
  DoubleTab& present = eqn.inconnue().valeurs(), &futur = eqn.inconnue().futur();
 
  // g1=futur=f(y0)
  DoubleTrav g1(present), g2(present); // just for initializing the array structure ...
 
  // sauv=y0
  DoubleTrav sauv(present);
  sauv = present;
 
  eqn.derivee_en_temps_inco(g1);
 
  // g1=hf(y0)
  g1 *= dt_;
 
  // present=y0+c2g1
  present.ajoute(c2, g1);
 
  // g2=futur=f(y0+c2g1)
  eqn.derivee_en_temps_inco(g2);
 
  // g2=b2"g2"
  g2 *= (b2 * dt_);
 
  // g2=b2"g2" + b1g1
  g2.ajoute(b1, g1);
  // g2.axpby(b2*dt_, g2, b1, g1); to fuse 2 kernels
 
  // normeg2=("g2","g2")
  double normeg2 = mp_carre_norme_vect(g2) + DMINFLOAT;
  // normeg1=-(g1,g1)
  double normeg1 = -mp_carre_norme_vect(g1);
  // psc1=2(g1,"g2")
  double psc1 = 2.0 * mp_prodscal(g1, g2);
 
  // y1=y0+(2g1(g1,"g2")-("g2")(g1,g1)/("g2","g2")
  // ToDo: implement axpby(a, x, b, y, result);
  //futur.axpby(psc1/(normeg2*dt), g1, normeg1, g2/(normeg2*dt)); to fuse 4 kernels
  futur = g1;
  futur *= psc1;
  futur.ajoute(normeg1, g2);
  futur /= (normeg2 * dt_);
  // futur = (g1 * psc1 + g2 * normeg1)/(normeg2 * dt_)
  present = sauv;
  update_critere_statio(futur, eqn);
  futur *= dt_;
  futur += sauv;
  // futur = sauv + (g1 * psc1 + g2 * normeg1)/normeg2
  eqn.domaine_Cl_dis().imposer_cond_lim(eqn.inconnue(), temps_courant() + pas_de_temps());
  futur.echange_espace_virtuel();
 
  return 1;
}